Appl Math Optim 53:311–330 (2006)
2006 Springer Science+Business Media, Inc.
Regularization of Nonmonotone Variational Inequalities
Igor V. Konnov,
M. S. S. Ali,
and E. O. Mazurkevich
Department of Applied Mathematics, Kazan University,
ul. Kremlevskaya 18, Kazan 420008, Russia
Department of Mathematics, Faculty of Education,
Ain Shams University, Cairo, Egypt
Informatics Problems Institute of AS RT,
Kazan 420012, Russia
Abstract. In this paper we extend the Tikhonov–Browder regularization scheme
from monotone to rather a general class of nonmonotone multivalued variational
inequalities. We show that their convergence conditions hold for some classes of
perfectly and nonperfectly competitive economic equilibrium problems.
Key Words. Multivalued variational inequalities, Regularization, Nonmonotone
mappings, Economic equilibrium problems.
AMS Classiﬁcation. Primary 49J20, Secondary 47H14, 47N10, 65K10, 91B50.
Let X be a nonempty convex set in the real n-dimensional space R
, and let G: X →
) be a multivalued mapping (here and below (A) denotes the family of all
nonempty subsets of a set A). Then one can deﬁne the multivalued variational inequality
problem (VI for short): Find a point x
∈ X such that
, x − x
≥0, ∀x ∈ X. (1)
In what follows we denote the solutions set of this problem by X
. Variational inequal-
ities are used for formulating and solving various equilibrium-type problems arising in
mathematical physics, economics, engineering and other ﬁelds. Most existing solution
methods for VIs require additional strict monotonicity-type assumptions for conver-
gence. One of the most popular approaches to solve nonstrictly monotone VIs is the